Tags: #projects/notes/reading #geomtop/symplectic-topology #geomtop/contact-topology #physics

Refs: Unsorted/Stein

Motivation

Example (Hypersurfaces of contact type): The level sets of a Hamiltonian on \({\mathbf{R}}^{2n} = \mathop{\mathrm{span}}_{\mathbf{R}}\left\{{\mathbf{p}, \mathbf{q}}\right\}\) given by \(H = K + U\) where \(K = \frac 1 2 {\left\lVert {\mathbf{p}} \right\rVert}^2\) and \(U = U(\mathbf{q})\) is a function of only \(\mathbf{q}\). (Usually kinetic + potential energy.)

Remark: all hypersurfaces of contact type \((X, \omega)\) look locally like \(X \hookrightarrow\mathrm{Sp}(X)\), i.e. \(X\) embedded into its symplectification.

Basic Questions:

• Basic question: when does the flow admit a periodic orbit?
• Does every/any vector field on a smooth manifold \(M\) admit a closed orbit?
  • Corollary: does every/any vector field on \(M\) admit a fixed point?
  • Note that if \(\chi(M) \neq 0\), the Poincare-Hopf index theorem forces every vector field to have a fixed point.
• Does every vector field on \(S^3\) admit a closed orbit?
  • Answer: no, very difficult to show, but turns out to hold for all 3-manifolds.

Remark: The orbit of a Hamiltonian flow is contained in a single level set.

Example: Simple Harmonic Oscillator.

• \(K = \frac 1 2 mv^2 = \frac{p^2}{2m}\) where \(p=mv\) is the momentum, given by \(F = ma\)
• \(U = \frac 1 2 kx^2\), given by Hooke’s law
• \(H(x, p) = U + K = \frac 1 2 mv^2 = \frac{p^2}{2m} + \frac 1 2 kx^2 \sim p^2 + x^2\)
• Has “phase space” \(\Phi = {\mathbf{R}}^2 = \mathop{\mathrm{span}}_{\mathbf{R}}\left\{{x, p}\right\}\), i.e. a position and momentum completely characterize the system at any fixed time.
• Conservation of energy shows that the time evolution of the system is governed by \({\frac{\partial x}{\partial t}\,} = -{\frac{\partial H}{\partial p}\,}\) and \({\frac{\partial p}{\partial t}\,} = {\frac{\partial H}{\partial x}\,}\)

  • Corresponds to a path \(\gamma: {\mathbf{R}}\to \Phi\) along which \(H\) is constant, i.e. a constant energy hypersurface corresponding (roughly) to \(p^2 + q^2 = \mathrm{const}\)

    • If the Hamiltonian evolved over time, this region would travel around phases space, with the volume of this region invariant.

Define Reeb flow

Define Reeb vector field